## References

SHOWING 1-10 OF 21 REFERENCES

### Quantitative Helly-Type Theorem for the Diameter of Convex Sets

- MathematicsDiscret. Comput. Geom.
- 2017

A version of the “quantitative” diameter theorem of Bárány, Katchalski and Pach, with a polynomial dependence on the dimension.

### Colorful Helly-type Theorems for Ellipsoids

- Mathematics
- 2019

We prove the following Helly-type result. Let $\mathcal{C}_1,\ldots,\mathcal{C}_{3d}$ be finite families of convex bodies in $\mathbb{R}^d$. Assume that for any colorful choice of $2d$ sets,…

### Brascamp-Lieb inequality and quantitative versions of Helly's theorem

- Mathematics
- 2015

We provide a number of new quantitative versions of Helly's theorem. For example, we show that for every family $\{P_i:i\in I\}$ of closed half-spaces $$P_i=\{x\in {\mathbb R}^n:\langle x,w_i\rangle…

### Quantitative combinatorial geometry for concave functions

- MathematicsJ. Comb. Theory, Ser. A
- 2021

### HELLY-TYPE THEOREMS AND GEOMETRIC TRANSVERSALS

- Mathematics
- 2016

INTRODUCTION Let F be a family of convex sets in R. A geometric transversal is an affine subspace that intersects every member of F . More specifically, for a given integer 0 ≤ k < d, a k-dimensional…

### Proof of a Conjecture of Bárány, Katchalski and Pach

- MathematicsDiscret. Comput. Geom.
- 2016

Bárány, Katchalski and Pach (Proc Am Math Soc 86(1):109–114, 1982) (see also Bárány et al., Am Math Mon 91(6):362–365, 1984) proved the following quantitative form of Helly’s theorem. If the…

### Polynomial estimates towards a sharp Helly-type theorem for the diameter of convex sets

- Mathematics
- 2018

We discuss a problem posed by Bárány, Katchalski and Pach: if {Pi : i ∈ I} is a family of closed convex sets in R such that diam (⋂ i∈I Pi ) = 1 then there exist s 6 2n and i1, . . . , is ∈ I such…

### Bounded boxes, Hausdorff distance, and a new proof of an interesting Helly-type theorem

- MathematicsSCG '94
- 1994

Two geometric optimization problems to convex programming are reduced: finding the largest axis-aligned box in the intersection of a family of convex sets and finding the translation and scaling that minimizes the Hausdorff distance between two polytopes.

### Quantitative Combinatorial Geometry for Continuous Parameters

- MathematicsDiscret. Comput. Geom.
- 2017

We prove variations of Carathéodory’s, Helly’s and Tverberg’s theorems where the sets involved are measured according to continuous functions such as the volume or diameter. Among our results, we…